I'm interested in studying Analytic Number Theory using Apostol's Introduction to Analytic Number Theory.
Having looked briefly at Apostol's Introduction to Analytic Number Theory and having read the reviews, I can see that complex analysis is assumed in this book.
In my degree I studied some complex analysis (up to and including the residue theorem). However this was things like using Laurent Series, contour integration methods and stuff - I'm not sure if that would count as complex 'analysis' (as this was not a hugely rigorous course (in the sense that a Real Analysis course would cover deriving the derivative and stuff by use of limits, convergence etc)).
Also I did not study any real analysis. I also studied a lot of pure maths (group theory, ring and field theory, number theory).
Would it be sufficient (or necessary) to go over:
Basic real analysis (up to convergence of functions). based on the book: http://www.springer.com/us/book/9781493927111 up to the chapter 'Functional Limits and Continuity' and
Complex Variable up to and including the residue theorem (based on this book: https://www.pearsonhighered.com/program/Osborne-Complex-Variables-and-their-Applications/PGM271956.html Chapters 1-5 (in particular, is this rigorous enough for a prerequisite to Analytic Number Theory? This was the book used when I was an undergraduate (in actual fact taught by the author)
Thank you for any guidance.
It's been a few years, but I once taught a course out of that book. As far as I can recall, the complex analysis you need is contour integration. I don't recall any "functional limits". There is a good bit of analyzing average values of arithmetic functions, so things like $F(n) =(1/n) \sum_{k=1}^n \sigma(k).$ But it's not much more complicated than the Series chapter in a calculus book. I think you're good to go.